Phase-resolved measurement and control of ultrafast dynamics in terahertz electronic oscillators

As a key component for next-generation wireless communications (6 G and beyond), terahertz (THz) electronic oscillators are being actively developed. Precise and dynamic phase control of ultrafast THz waveforms is essential for high-speed beam steering and high-capacity data transmission. However, measurement and control of such ultrafast dynamic process is beyond the scope of electronics due to the limited bandwidth of the electronic equipment. Here we surpass this limit by applying photonic technology. Using a femtosecond laser, we generate offset-free THz pulses to phase-lock the electronic oscillators based on resonant tunneling diode. This enables us to perform phase-resolved measurement of the emitted THz electric field waveform in time-domain with sub-cycle time resolution. Ultrafast dynamic response such as anti-phase locking behaviour is observed, which is distinct from in-phase stimulated emission observed in laser oscillators. We also show that the dynamics follows the universal synchronization theory for limit cycle oscillators. This provides a basic guideline for dynamic phase control of THz electronic oscillators, enabling many key performance indicators to be achieved in the new era of 6 G and beyond.

Terahertz (THz) electronics is a rapidly evolving research field 4 .THz wave emitters based on compact electronic oscillators are suitable for real-world applications such as THz imaging 9 , Radar 10 , and wireless communications towards 6G and beyond [1][2][3] .Several types of THz oscillators are developed using diodes 5,[11][12][13] and transistors [14][15][16] .They use negative differential conductance (NDC) as an alternating-current gain to convert direct-current power to continuous wave (CW) THz radiation.The modern microfabrication technology was very successful in reducing the inductance (L) and capacitance (C) of the device, leading to the ultrahigh operating frequency ( != 1 √ ⁄ ) of THz.The highest fundamental oscillation approaching 2 THz is achieved 17 by resonant tunneling diode (RTD) oscillator [18][19][20][21][22][23] .However, the characterization of such ultrafast oscillation is extremely challenging.For a clear example, THz frequency far exceeds the bandwidth of the state-of-the-art oscilloscopes, hindering the direct monitoring of the oscillation waveform 24 .
At the same time, however, ultrafast time-domain measurement of the THz waveform is now routinely performed in the field of THz photonics.Instead of the electronic means, this technique is based on optical sampling such as electro-optic sampling 25 using femtosecond lasers and known as THz time-domain spectroscopy [26][27][28][29] (THz-TDS).In THz-TDS, a time resolution only limited by the pulse width of the femtosecond laser is achieved by utilizing phase-stable periodic nature of the signal waveform.Although THz-TDS is an ideal tool for characterizing THz electronic oscillators, no successful measurement has been reported so far due to their random phase fluctuation.
In this paper, we show THz-TDS technique can be used to phase-lock electronic oscillators based on RTD by injection locking and perform optical sampling of the emitted THz waveform with a time resolution unachievable with electronic means.The phase-resolved measurement revealed the anti-phase locking nature, which is reproduced by the general synchronization theory for limit cycle (Van der Pol) oscillators.We also demonstrated oscillation phase control through the injection signal waveform.Our findings will provide the foundation for phase control technology of THz electronic oscillators.

Results
Principle of optical sampling.In order to perform optical sampling like THz-TDS, the signal waveform must be repetitive.In THz-TDS, this condition is satisfied because carrier-envelope phase stable THz pulses are generated by a mode-locked femtosecond laser 31,31 .In the frequencydomain, this is equivalent to the offset-free comb spectrum at nfrep (n: integer, frep : laser repetition rate) 32 .In this case, THz wave with a period of TTHz = (nfrep) -1 and sampling pulses derived from the same mode-locked laser with a repetition interval of Trep = (frep) -1 are phase-locked, i.e., Trep = nTTHz.Therefore, all the sampling pulses measure the same electric field value at a fixed phase (black solid circles on red solid waveforms in Fig. 1a) and this allows us to accumulate huge number of tiny signals to sufficiently suppress the noise.By changing the relative timing between the THz and sampling pulse, we can reconstruct the whole waveform of the THz radiation.
Similarly, time-domain sampling of the CW THz wave from electronic oscillators is possible if the oscillation frequency matches nfrep.However, this is generally unrealistic due to the phase fluctuation.In the case of RTD oscillators, typical phase coherence time is less than one microsecond 33 (spectral linewidth on the order of a few MHz).In this situation, sampling pulses detect random electric field values every one microsecond (schematically shown by open circles on dashed red waveforms in Fig. 1a) and the required accumulation process produces zero signal.
Therefore, phase locking is a key to realize optical sampling of electronic oscillators.
Recent research revealed that THz RTD oscillators can be phase-locked by injecting external THz wave with the amplitude as small as 10 -4 of the RTD oscillator itself 33 .For typical RTD oscillators with 10 µW output power, this corresponds to the electric field amplitude on the order of mV/cm, which is achievable by THz pulses used in THz-TDS.In this study, we used offsetfree THz pulses with a repetition rate of frep for injection locking (Fig. 1b).This allows us to lock the oscillation frequency and phase of the RTD oscillator to nfrep and satisfy the requirement for the optical sampling.
Time-domain measurement.We used an RTD oscillator made of AlAs/GaInAs double barrier structure 34 (see Methods and Supplementary Fig. S1 for detail).The size of the RTD chip is very tiny, comparable to a laser diode.The experimental setup is a standard THz-TDS system in reflection geometry (see Methods and Supplementary Fig. S2 for detail).We injected THz pulses to the biased RTD and measured reflected THz wave (Eon) using electro-optic sampling technique.
If the RTD oscillation is phase-locked by the THz pulse, we should be able to observe the electric field emitted by the RTD in addition to the reflected injection THz pulse.By subtracting the waveform of the injection THz pulse which can be measured without the bias voltage (Eoff), we extracted the emission from the RTD (ERTD = Eon -Eoff).To avoid long-term fluctuation, we employed double modulation technique and obtained Eoff and ERTD simultaneously in a single delay stage scan rather than performing two scans for Eon and Eoff (see Methods).
Figures 1c and 1d show the electric field waveforms of the injection THz pulse (Eoff) and the difference signal (ERTD), respectively.The bias voltage was set within the oscillation region (510 mV).Before the injection THz pulse comes at 9 ps, ERTD only shows noisy signal although the RTD is emitting THz wave.This is the consequence of the phase fluctuation described above.
After the THz pulse injection, the difference signal starts to show sinusoidal oscillation with a frequency of 0.340 THz.This agrees with the free-running frequency and suggests the successful phase-locking and phase-resolved detection of the RTD emission in time-domain.For the further confirmation, we calculated the Fourier transform ( ) on and  ) off ) and checked the amplitude ratio (* ) on * * ) off * + ).As shown by the red curve in Fig. 1g, a sharp peak above one at 0.340 THz is seen, which means that the oscillation in Fig. 1c is due to the RTD emission, eliminating the possibility of absorption signal.
Figure 1e shows the difference signal measured when the RTD was biased just outside of the oscillation region (551 mV).Even though the RTD was not oscillating, we still observed finite signal.This is because the reflectivity of the RTD slightly changes by the bias voltage most likely due to the rectification by the RTD as a THz detector 35,36 .The sudden increase of the signal around Although a small dip around 346.5 GHz is unavoidable, the large signal at 35 ps disappears and smooth buildup of the phase-locked component in about 50 ps is seen.For the decay dynamics, we don't have to rely on the numerical data processing because the signal due to the RTD reflectivity change becomes negligible after 80 ps (Fig. 1e).After 80 ps, the RTD emission amplitude gradually decreases and becomes almost constant after around 250 ps (for full waveform up to 600 ps, see Fig. 2a).
The electric field measurement in time-domain allows us to determine the phase of the RTD oscillation.We fitted the RTD emission after 80 ps by a decaying cosine function with its time origin at 9 ps and obtained the initial phase of 1.10p.The phase of the injection signal at 0.340 THz is also obtained (0.11p) from the Fourier transform using cosine functions starting from 9 ps (inset of Fig. 1c).We found that the phase difference is 0.99p, which means that the RTD is antiphase locked to the injection signal.These dynamical features will be explained by a general synchronization theory for limit cycle oscillators.

Bias voltage dependence. We measured differential signals at different bias voltages to see how
the RTD emission changes.Typical waveforms are shown in Fig. 2a (see Supplementary Fig. S3 for the complete data set).Similar dynamics as discussed above is observed for all bias voltages, but with different oscillation frequencies and amplitudes.Figure 2b shows the power spectrum (* ) RTD * ( ) calculated from the time-domain data after 80 ps.The oscillation frequency determined from the peak position changes with the bias voltage as shown by the red circles in Fig. 2c, and follows the free-running oscillation frequency (blue trace).The intensity of the RTD emission component was evaluated from the peak area of the power spectrum.As plotted in Fig. 2d by the red circles, the overall trend is similar to the output intensity in the free-running state (blue) especially above 480 mV.The fluctuation below 480 mV is probably due to the interference effect that increases or decreases the signal as the oscillation frequency changes with the bias voltage.
The phase shift of the RTD oscillation with respect to the injection pulse is summarized in Fig. 2e.Although it shows some deviation, again below 480 mV, the data points are concentrated around p irrespective of the bias voltage.Effect of phase fluctuation.Although the electric field amplitude is stable up to 600 ps (Fig. 2a), it actually decays and becomes almost zero before 12.3 ns when the next injection THz pulse comes (at 81 MHz repetition rate).This is evident from the noise signal before the THz pulse injection shown in Fig. 1d.In this experiment, the differential signal decays due to the phase fluctuation.Therefore, the experimental results mean that the phase coherence time is shorter than 12.3 ns, which corresponds to the linewidth broader than 81 MHz.This is much broader than the typical linewidth of a few MHz 33 , but can happen due to the noise from the power supply (in the current experiment, a function generator for double modulation measurement).
In some cases, however, we observed differential signals with longer coherence times than 12.3 ns. Figure 3 shows one example observed at 485 mV.We can see oscillation signal before the THz pulse injection, which is actually the RTD emission signal phase-locked by the previous injection THz pulses.This linewidth narrowing can be attributed to the effect of the cavity 33 formed by the RTD and optical components such as electro-optic crystal or photo-conductive antenna (see Supplementary Fig. S2a).This happens only when the RTD oscillation frequency coincides with one of the longitudinal modes of the cavity.Analysis based on Van der Pol model.To understand the ultrafast dynamics of the injection locking, we performed numerical simulations and theoretical analysis based on a simple equivalent circuit 39 (LC oscillator) shown in Fig. 4a.Here, v(t) is the oscillation voltage, i(v) is the alternating current flowing through the RTD.For the current-voltage characteristics of the RTD, we assumed a cubic function, i(v) = -av + gv 3 , where a and g are positive constants (Fig. 4b).GL is the load conductance of the antenna that accounts for the radiation loss and is equal to a/2 for the maximum radiation power 40 .The injection signal (vinj) is modelled as a voltage source connected in series with the antenna conductance.The circuit equation becomes Van der Pol equation with external forcing term, which describes synchronization (injection locking) phenomena 41 .
Here,  = 46  ⁄  is the dimensionless oscillation amplitude, ) is the dimensionless time, and  = 4  ⁄ 2 ⁄ is the nonlinearity parameter.The over-dot denotes the derivative with respect to .
As shown below, the time scale of the signal decay allows us to estimate the value of  as around 0.01.This enables us to simulate the RTD oscillator without assuming specific values for the circuit parameters (C, L, a and g).To simulate the typical situation with a short coherence time (Figs. 1 and 2), we only considered a single injection pulse.This is because the oscillation phase changes in a random fashion before the next injection pulse comes and cumulative effect does not exist.To mimic the phase fluctuation and data accumulation process in the experiment, we numerically solved eq. ( 1) with twenty different initial phases and took the average of them (〈〉).
Figure 4d shows the simulated injection locking dynamics (〈〉, blue trace, left axis) induced by the injection pulse (orange trace, right axis).We assumed multi-cycle injection waveform to simulate the frequency filtering effect due to the antenna and circuit (see Methods for detail).The bottom axis is the dimensionless time  and the top axis is the corresponding time when  !2 ⁄ = 340 GHz.After the zero-signal due to the phase fluctuation, we see finite signal reflecting the phase-locking effect.The same waveform repeats at the rate of frep, which means that its frequency spectrum consists of comb modes at nfrep (i.e., injection locking to nfrep).The build-up time of the phase-locked signal is around 50 ps, which reproduces the experimental results (Fig. 1f).This result shows that the build-up time is determined by the duration of the injection signal.After the signal reaches its maximum, the amplitude gradually decreases and becomes stable after ~ 250 ps, as observed in the experiment (Fig. 2a).The decay time here strongly depends on the value of , which allowed us to estimate its value.Finally, if we look at the oscillation phase (Fig. 4e), this model reproduces the anti-phase locking behaviour observed in the experiment (Fig. 2e).
The anti-phase locking nature can be explained by the phase response function of the forced Van der Pol oscillator (see Methods for derivation), Here  inj,0 > 0 and  0 > 0 are the amplitudes of the injection signal and the oscillator, respectively.() is the phase shift between the injection signal and the oscillator.In the steady state (̇= 0), we have only one stable solution at  =  (Fig. 4c), which means the oscillator shows anti-phase locking behaviour.Oscillation phase control.The forced Van der Pol model predicts a constant phase shift between the injection signal and the electronic oscillator.This allows us to arbitrarily control the oscillation phase by changing the phase of the injection signal.For a simple demonstration, we performed a similar experiment as Fig. 1d, but with polarity-reversed (p phase shifted) injection THz pulse.
The red trace in Fig. 5 shows the same data as plotted in Fig. 1d, while the black trace shows the data taken with polarity-reversed injection THz pulse.As expected, the oscillation phase of the RTD emission is also reversed.

Discussion and outlook
Phase control of THz electronic oscillators is an indispensable prerequisite for realizing many functionalities and applications such as beam forming/steering 42 , radar and high-capacity wireless communications.Injection-locked RTD THz oscillators are promising devices for such operation because they allow phase control by electronic signal (bias voltage) 43 .This gives hope that highspeed phase modulation on the order of tens of GHz will be possible.However, the corresponding modulation period reaches as short as several tens of picoseconds, which is comparable to the time scale of the transient dynamics observed in this paper.This indicates that the maximum modulation bandwidth will be limited by the dynamical response time.This also shows the significance of understanding and controlling the ultrafast dynamics of the injection-locked THz oscillator.At the same time, ultrafast time-domain measurement of the THz electronic oscillators in the free-running state is also important to understand their phase fluctuation characteristics.For such purposes, single-shot THz TDS technique 44,45 that can capture non-repetitive waveform is necessary instead of the method presented in this paper.
It is worth comparing with similar optical sampling measurements performed in THz quantum cascade lasers (QCLs) 6,7 .In QCLs, offset-free THz pulses are injected and used as seed light for lasing via stimulated emission.Consequently, the emitted THz wave has the same phase as the THz pulse 8 .The in-phase emission is a hallmark of the laser oscillator.In contrast, the anti-phase locking behaviour observed in RTD reflects the nature of the electronic oscillator.Although THz QCLs and RTDs have much in common 46,47 , our results show that the locking dynamics is clearly different.
In conclusion, we showed that optical sampling technique is applicable to THz electronic oscillators by injection locking.This phase-resolved measurement allows us to directly monitor the ultrafast oscillation dynamics in response to external signal, which will be of significant importance for future applications such as ultrafast modulation in THz wireless communication.
Based on the successful modeling using Van der Pol oscillator, we also demonstrated oscillation phase control through the phase of the injection waveform.This method will be universally applicable to other types of electronic limit cycle oscillators and provide a new tool for characterizing and controlling ultrafast response of THz electronic oscillators.
35 ps is explained by the internal reflection inside the device (26 ps round trip time.See Supplementary information for detail).Similar increase at 35 ps in Fig.1dsuggests that the signal due to the RTD reflectivity change is superimposed on the RTD emission.This point can be confirmed by comparing the amplitude ratio for both voltages (red and black traces in Fig.1g); except for the RTD emission peak, they have small peaks and dips in common.These structures are almost equally spaced by 38.5 GHz (as indicated by the vertical grids), which is consistent with the existence of the multiple internal reflection of a round-trip time of 26 ps (= 38.5 GHz -1 ).Injection locking dynamics.By eliminating the signal due to the RTD reflectivity change, we can extract the RTD emission component from the difference signal.The narrow spectrum of the RTD emission allows us to use a simple frequency filtering to suppress the signal due to the reflectivity change.We numerically filtered out the frequency component outside of the blue shaded region in Fig.1gand extracted the RTD emission component (blue trace in Fig.1f).

Figure 1 |
Figure 1 | Phase-resolved time-domain sampling of RTD THz oscillator.a. Principle of optical sampling.THz wave (solid red curve) and sampling pulses are phase-locked.Dashed red curve represents the THz wave from free-running electronic oscillators with phase noise.b.Phaselocking of RTD by offset-free THz pulse injection.c.Waveform of the injection THz pulse (Eoff).The inset shows the amplitude (orange) and phase (blue) in the frequency-domain calculated from Eoff in the first 18 ps.The time origin was taken at 9 ps (positive peak of the main pulse) for the phase determination based on cosine function.Small-amplitude pulses at 22 ps and 94 ps are due to internal reflections inside the silicon substrate and detection crystal, respectively.d.Waveform of the difference signal (ERTD) at 510 mV.e. Waveform of the difference signal at 551 mV which is just outside of the oscillation region.f.Waveform of the RTD emission component (blue curve) at 510 mV obtained by frequency filtering.g.Fourier amplitude ratio between bias voltage on and off (red: 510 mV, black: 551 mV).The vertical grid lines are drawn at the integer multiples of 38.5 GHz.The frequency resolution is 1.67 GHz.The amplitude ratio is almost unity in other frequency region.Only the spectral components in the blue shaded region are used to calculate the blue curve in d.

Figure 2 |
Figure 2 | Bias voltage dependence.a.Typical waveforms and b. power spectra of the differential signals at three different bias voltages.To exclude the effect of the RTD reflectivity change, we used time-domain data after 80 ps to obtain the power spectra.c.Oscillation frequencies determined from the peak position of the power spectra (red circles).The frequency resolution is 1.92 GHz.Blue curve represents the oscillation frequency in the free-running state determined by conventional heterodyne down-conversion method 37 .d. Output intensity determined from the peak area of the power spectrum (red circles).Blue curve is the output intensity in the free-running state measured with a square law detector (Fermi-level managed barrier diode 38 ).The data point at 485 mV is omitted due to the different situation of the phase-locking as shown in Fig. 3. e. Phase shift determined from the electric field oscillation.

Figure 3 |
Figure 3 | RTD emission with long coherence time.Oscillation signal is observed before the injection THz pulse arrived at 9 ps.The signal intensity is much stronger than those shown in Fig. 1 due to the cumulative effect by multiple injection THz pulses.

Figure 4 |
Figure 4 | Analysis based on Van der Pol model.a. Equivalent circuit.GL: load conductance, C: capacitance, L: inductance, i(v): RTD current, v(t): oscillation voltage.b.Current-voltage characteristic approximated by a cubic function.We considered the simplest case of the bias voltage at the center of the NDC region 35 (inflection point), but similar results can be obtained at other bias voltages.c.Phase response function of the forced Van der Pol oscillator, showing the stable steady-state solution at p. d.Simulated waveform (blue, left axis) phase-locked by the injection signal (orange, right axis).e. Magnified view of d, showing the anti-phase locking behaviour.

Figure 5 |
Figure 5 | Oscillation phase control.The polarity of the injection THz pulse is reversed by flipping the sign of the voltage applied to the THz pulse emitter (photo-conductive antenna).

Figure S1 |
Figure S1 | Current-voltage and output intensity-voltage characteristics of RTD THz oscillator.The RTD was driven by a source meter in voltage source mode.The output intensity is the same one shown in Fig.2das blue curve.

Figure S2 |
Figure S2 | Experimental setup.a. THz-TDS setup in reflection geometry.b.Timing chart for the double modulation.

Figure S3|
Figure S3| Bias voltage dependence of the difference signal ERTD.The bias voltages of the typical waveforms used in the main text are underlined.